I would say it's a mesembryanthemum of the genus Faucaria - I would only be guessing at the species. The flowers are yellow when fully open but in most species the petals turn dark pink or red as the flower dies.
They are mostly native to South Africa and I think the Americans call them the Tiger's Jaw mesems.
Nice photo.
Bruce
ps: So not really a member of the cactus family (Cactaceae).
I agree with Bruce. Your first image is in the faucaria family. Your second image is also a mesemb, but it is in the aloinopsis genus. I like the second image best.
Macroluv. I am cactus only so not into other succulents but would agree with the other two xerophiles above!
The only cacti that look similar to your plants are the Ariocarpus group, which is perhaps what you have seen in the past and are confusing them with? See:-